Subgroups — C₂×#13131192 #16807036

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88104	C₂	1	1	yes	maximal, characteristic
2	C₂×D₁₃⋊#108	C₂	1	1	yes	maximal, characteristic
3	#13131192	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718224	C₂	1	2	yes	maximal, complemented
5	C₂×#11718223	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
7	C₂×D₁₃⋊M₁₆	—	2	1	no	—
8	C₂²×#6118	—	2	1	no	—
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	D₁₃⋊#108	—	2	2	no	—
11	D₁₃⋊#108	C₂²	1	4	yes	—
12	C₂×#88104	—	2	1	no	—
13	C₂×#88104	—	2	2	no	—
14	C₂×#88104	C₂²	1	1	yes	characteristic
15	C₂×#88104	C₂²	1	2	yes	—
16	C₂×#88104	C₂²	1	1	yes	characteristic
17	C₂×#88104	C₂²	1	4	yes	—
18	C₂×#88103	—	2	1	no	—
19	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic
20	C₂×#88052	C₄	1	1	yes	characteristic
21	D₁₃×M₃₂	C₄	1	4	yes	—
22	D₁₃×M₃₂	C₂²	1	4	yes	—
23	C₂×#88038	C₄	1	2	yes	—
24	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
25	#11718224	C₂²	1	8	yes	complemented
26	#11718223	C₂²	1	8	yes	complemented
27	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
28	C₂×#6122	—	4	1	no	—
29	C₂²×#1057	—	4	1	no	—
30	D₁₃⋊M₁₆	—	2	4	no	—
31	D₁₃⋊M₁₆	—	4	2	no	—
32	C₂×#6118	—	2	1	no	—
33	C₂×#6118	—	2	1	no	—
34	C₂×#6118	—	2	2	no	—
35	C₂×#6118	—	2	4	no	—
36	C₂×#6118	—	4	1	no	—
37	C₂×#6118	—	4	2	no	—
38	C₂×D₁₃⋊C₈	—	4	1	no	—
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×#1055	Q₈	1	1	yes	characteristic
41	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
44	C₂×C₈×D₁₃	D₄	1	2	yes	—
45	C₂×C₈×D₁₃	Q₈	1	2	yes	—
46	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
47	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₁₃	C₂³	1	2	yes	—
49	#88104	—	2	4	no	—
50	#88104	—	2	2	no	—
51	#88104	—	2	1	no	—
52	#88104	—	2	1	no	—
53	#88104	C₂³	1	2	yes	—
54	#88104	C₂³	1	4	yes	—
55	#88104	C₂³	1	2	yes	—
56	#88103	—	2	4	no	—
57	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
58	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
59	#88052	C₂×C₄	1	4	yes	—
60	C₂×#6090	C₂×C₄	1	2	yes	—
61	#88038	C₂×C₄	1	8	yes	—
62	C₁₆×D₁₃	C₂×C₄	1	4	yes	—
63	C₁₆×D₁₃	C₂³	1	4	yes	—
64	C₂×#2161	—	13	1	no	maximal, Sylow
65	C₂³×D₁₃	Q₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
68	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
71	C₂×C₄×D₁₃	D₈	1	2	yes	—
72	C₂×C₄×D₁₃	Q₁₆	1	1	yes	characteristic
73	C₂×C₄×D₁₃	Q₁₆	1	2	yes	—
74	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
76	#6122	—	4	4	no	—
77	C₂×#1057	—	4	1	no	—
78	C₂×#1057	—	4	1	no	—
79	C₂×#1057	—	4	2	no	—
80	C₂×#1057	—	4	4	no	—
81	#6118	—	2	2	no	—
82	#6118	—	2	2	no	—
83	#6118	—	2	4	no	—
84	#6118	—	4	4	no	—
85	#6118	—	4	2	no	—
86	#6118	—	4	1	no	—
87	#6118	—	4	1	no	—
88	D₁₃⋊C₈	—	4	4	no	—
89	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
94	C₂×#1055	C₄⋊C₄	1	2	yes	—
95	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
96	C₂×#1055	C₂×Q₈	1	2	yes	—
97	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
98	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
99	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
100	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₈×D₁₃	C₂²×C₄	1	2	yes	—
102	C₈×D₁₃	C₂×D₄	1	2	yes	—
103	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
104	C₈×D₁₃	C₂×Q₈	1	1	yes	characteristic
105	C₈×D₁₃	C₂×Q₈	1	2	yes	—
106	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
107	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
108	#6090	C₂²×C₄	1	4	yes	—
109	C₂²×M₃₂	—	13	1	no	—
110	C₂²×#108	—	13	1	no	—
111	C₂²×#108	—	13	1	no	—
112	#2161	—	13	8	no	—
113	C₂×#368	—	13	2	no	—
114	C₂×#368	—	13	2	no	—
115	C₂³×C₁₃	#107	1	1	yes	characteristic
116	C₂²×D₁₃	—	4	2	no	—
117	C₂²×D₁₃	#107	1	1	yes	characteristic
118	C₂²×D₁₃	#107	1	2	yes	—
119	C₂²×D₁₃	C₂×Q₁₆	1	1	yes	characteristic
120	C₂²×D₁₃	C₂×Q₁₆	1	2	yes	—
121	C₂×C₄×C₁₃	#107	1	2	yes	—
122	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×Dic₁₃	#107	1	2	yes	—
126	C₂×Dic₁₃	#107	1	1	yes	characteristic
127	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
128	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
129	C₄×D₁₃	#107	1	2	yes	—
130	C₄×D₁₃	#107	1	2	yes	—
131	C₄×D₁₃	#107	1	2	yes	—
132	C₄×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₄×D₁₃	C₂×D₈	1	2	yes	—
135	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
136	C₄×D₁₃	C₂×Q₁₆	1	1	yes	characteristic
137	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
138	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
139	#1057	—	4	2	no	—
140	#1057	—	4	2	no	—
141	#1057	—	4	4	no	—
142	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
143	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
145	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
146	#1055	C₂×C₄⋊C₄	1	2	yes	—
147	#1055	C₂²×Q₈	1	1	yes	characteristic
148	C₂²×M₁₆	—	26	1	no	—
149	C₂²×M₁₆	—	26	1	no	—
150	C₂³×C₈	—	13	1	no	—
151	C₂×M₃₂	—	13	1	no	—
152	C₂×M₃₂	—	13	4	no	—
153	C₂×M₃₂	—	13	4	no	—
154	C₂×M₃₂	—	13	2	no	—
155	C₂×M₃₂	#261	1	1	yes	characteristic
156	C₂²×C₁₆	—	13	2	no	—
157	C₂×#108	—	13	1	no	—
158	C₂×#108	—	13	1	no	—
159	C₂×#108	—	13	2	no	—
160	C₂×#108	—	13	4	no	—
161	C₂×#108	—	13	4	no	—
162	C₂×#108	—	26	1	no	—
163	C₂×#108	—	26	2	no	—
164	C₂×#108	—	26	1	no	—
165	C₂×#108	—	26	2	no	—
166	#368	—	13	8	no	—
167	#368	—	13	8	no	—
168	C₂²×C₁₃	—	2	2	no	—
169	C₂²×C₁₃	C₂×#107	1	2	yes	—
170	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
171	C₂×D₁₃	—	2	2	no	—
172	C₂×D₁₃	—	2	2	no	—
173	C₂×D₁₃	—	4	2	no	—
174	C₂×D₁₃	—	4	2	no	—
175	C₂×D₁₃	C₂×#107	1	2	yes	—
176	C₂×D₁₃	C₂×#107	1	1	yes	characteristic
177	C₂×D₁₃	C₂²×Q₁₆	1	1	yes	characteristic
178	C₄×C₁₃	C₂×#107	1	2	yes	—
179	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
180	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
181	Dic₁₃	C₂×#107	1	1	yes	characteristic
182	Dic₁₃	C₂×#107	1	2	yes	—
183	Dic₁₃	C₂²×D₈	1	1	yes	characteristic
184	C₂³×C₄	—	13	1	no	—
185	C₂×M₁₆	—	26	2	no	—
186	C₂×M₁₆	—	26	4	no	—
187	C₂×M₁₆	—	26	1	no	—
188	C₂×M₁₆	—	26	1	no	—
189	C₂×M₁₆	—	26	4	no	—
190	C₂×M₁₆	—	52	1	no	—
191	C₂×M₁₆	—	52	2	no	—
192	C₂×M₁₆	—	52	2	no	—
193	C₂×M₁₆	—	52	1	no	—
194	C₂²×C₈	—	13	1	no	—
195	C₂²×C₈	—	13	1	no	—
196	C₂²×C₈	—	13	2	no	—
197	C₂²×C₈	—	13	2	no	—
198	C₂²×C₈	—	13	1	no	—
199	C₂²×C₈	—	13	2	no	—
200	C₂²×C₈	—	13	1	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₂²×C₈	—	13	2	no	—
203	C₂²×C₈	—	52	1	no	—
204	C₂²×C₈	—	52	1	no	—
205	C₂²×C₈	C₂×#261	1	1	yes	characteristic
206	M₃₂	—	13	4	no	—
207	M₃₂	—	13	8	no	—
208	M₃₂	C₂×#261	1	4	yes	—
209	C₂×C₁₆	—	13	2	no	—
210	C₂×C₁₆	—	13	4	no	—
211	C₂×C₁₆	—	13	4	no	—
212	C₂×C₁₆	C₂×#261	1	2	yes	—
213	#108	—	13	2	no	—
214	#108	—	13	4	no	—
215	#108	—	13	2	no	—
216	#108	—	26	4	no	—
217	#108	—	26	2	no	—
218	#108	—	26	4	no	—
219	#108	—	26	1	no	—
220	#108	—	26	1	no	—
221	C₂×C₁₃	—	2	2	no	—
222	C₂×C₁₃	#2161	1	2	yes	complemented
223	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
224	D₁₃	—	2	1	no	—
225	D₁₃	—	2	2	no	—
226	D₁₃	—	2	1	no	—
227	C₂⁴	—	13	1	no	—
228	C₂²×C₄	—	13	1	no	—
229	C₂²×C₄	—	13	1	no	—
230	C₂²×C₄	—	13	2	no	—
231	C₂²×C₄	—	13	2	no	—
232	C₂²×C₄	—	13	1	no	—
233	C₂²×C₄	—	13	2	no	—
234	C₂²×C₄	—	13	1	no	—
235	C₂²×C₄	—	13	2	no	—
236	C₂²×C₄	—	13	1	no	—
237	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
238	M₁₆	—	26	2	no	—
239	M₁₆	—	26	2	no	—
240	M₁₆	—	26	4	no	—
241	M₁₆	—	52	4	no	—
242	M₁₆	—	52	2	no	—
243	M₁₆	—	52	1	no	—
244	M₁₆	—	52	1	no	—
245	M₁₆	—	52	4	no	—
246	C₂×C₈	—	13	2	no	—
247	C₂×C₈	—	13	1	no	—
248	C₂×C₈	—	13	2	no	—
249	C₂×C₈	—	13	1	no	—
250	C₂×C₈	—	13	2	no	—
251	C₂×C₈	—	13	1	no	—
252	C₂×C₈	—	13	2	no	—
253	C₂×C₈	—	13	2	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	1	no	—
256	C₂×C₈	—	13	2	no	—
257	C₂×C₈	—	13	1	no	—
258	C₂×C₈	—	13	1	no	—
259	C₂×C₈	—	13	2	no	—
260	C₂×C₈	—	52	4	no	—
261	C₂×C₈	—	52	4	no	—
262	C₂×C₈	—	52	1	no	—
263	C₂×C₈	—	52	1	no	—
264	C₂×C₈	—	52	2	no	—
265	C₂×C₈	C₄⋊#261	1	2	yes	—
266	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
267	C₂×C₈	C₂²×#261	1	1	yes	characteristic
268	C₂×C₈	C₂²×#261	1	2	yes	—
269	C₁₆	—	13	4	no	—
270	C₁₆	C₂²×#261	1	4	yes	—
271	C₁₃	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
272	C₂³	—	13	1	no	—
273	C₂³	—	13	1	no	—
274	C₂³	—	13	2	no	—
275	C₂³	—	13	2	no	—
276	C₂³	—	52	2	no	—
277	C₂³	#88102	1	1	yes	characteristic
278	C₂×C₄	—	13	2	no	—
279	C₂×C₄	—	13	2	no	—
280	C₂×C₄	—	13	1	no	—
281	C₂×C₄	—	13	1	no	—
282	C₂×C₄	—	13	2	no	—
283	C₂×C₄	—	13	1	no	—
284	C₂×C₄	—	13	2	no	—
285	C₂×C₄	—	13	2	no	—
286	C₂×C₄	—	13	2	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	1	no	—
289	C₂×C₄	—	13	1	no	—
290	C₂×C₄	—	13	1	no	—
291	C₂×C₄	—	13	2	no	—
292	C₂×C₄	#88102	1	2	yes	—
293	C₂×C₄	#88102	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
295	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
296	C₈	—	13	1	no	—
297	C₈	—	13	2	no	—
298	C₈	—	13	1	no	—
299	C₈	—	52	2	no	—
300	C₈	—	52	2	no	—
301	C₈	—	52	4	no	—
302	C₈	C₂×C₄⋊#261	1	1	yes	characteristic
303	C₈	C₂×C₄⋊#261	1	2	yes	—
304	C₈	C₂³×#261	1	1	yes	characteristic
305	C₂²	—	2	2	no	—
306	C₂²	—	13	1	no	—
307	C₂²	—	13	2	no	—
308	C₂²	—	13	1	no	—
309	C₂²	—	26	2	no	—
310	C₂²	—	26	2	no	—
311	C₂²	—	52	2	no	—
312	C₂²	—	52	2	no	—
313	C₂²	C₂×#88102	1	2	yes	—
314	C₂²	C₂×#88102	1	1	yes	characteristic
315	C₄	—	13	1	no	—
316	C₄	—	13	2	no	—
317	C₄	—	13	1	no	—
318	C₄	C₂×#88102	1	2	yes	—
319	C₄	C₂×#88102	1	1	yes	characteristic
320	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
321	C₂	—	2	2	no	—
322	C₂	—	26	1	no	—
323	C₂	—	26	2	no	—
324	C₂	—	26	1	no	—
325	C₂	#13131192	1	2	yes	direct product factor
326	C₂	C₂²×#88102	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88102
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₈	C₂³×#261
Fitting	C₂×C₁₃×M₃₂	C₄
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×D₁₃	Q₁₆
cosocle	C₈×D₁₃	C₂⁴
hypercenter	C₂×M₃₂	#261
hypocenter	C₁₃	C₂×#2161
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #13131192	C₂	direct (N × H)
4. C₂×#11718224	C₂	semidirect
5. C₂×#11718223	C₂	semidirect
25. #11718224	C₂²	semidirect
26. #11718223	C₂²	semidirect
222. C₂×C₁₃	#2161	semidirect
271. C₁₃	C₂×#2161	semidirect
325. C₂	#13131192	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#88104	2
C₂×D₁₃⋊#108	2
#13131192	2
C₂×#11718224	2
C₂×#11718223	2
C₂×D₁₃×M₃₂	2
C₂×#2161	13

Subgroups — C2×#13131192 #16807036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131192 #16807036